3.8.61 \(\int \frac {(a+b x)^3}{(a^2-b^2 x^2)^2} \, dx\) [761]

Optimal. Leaf size=26 \[ \frac {2 a}{b (a-b x)}+\frac {\log (a-b x)}{b} \]

[Out]

2*a/b/(-b*x+a)+ln(-b*x+a)/b

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641, 45} \begin {gather*} \frac {2 a}{b (a-b x)}+\frac {\log (a-b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/(a^2 - b^2*x^2)^2,x]

[Out]

(2*a)/(b*(a - b*x)) + Log[a - b*x]/b

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rubi steps

\begin {align*} \int \frac {(a+b x)^3}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {a+b x}{(a-b x)^2} \, dx\\ &=\int \left (\frac {2 a}{(a-b x)^2}+\frac {1}{-a+b x}\right ) \, dx\\ &=\frac {2 a}{b (a-b x)}+\frac {\log (a-b x)}{b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 23, normalized size = 0.88 \begin {gather*} \frac {\frac {2 a}{a-b x}+\log (a-b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/(a^2 - b^2*x^2)^2,x]

[Out]

((2*a)/(a - b*x) + Log[a - b*x])/b

________________________________________________________________________________________

Maple [A]
time = 0.44, size = 27, normalized size = 1.04

method	result	size
default	\(\frac {2 a}{b \left (-b x +a \right )}+\frac {\ln \left (-b x +a \right )}{b}\)	\(27\)
risch	\(\frac {2 a}{b \left (-b x +a \right )}+\frac {\ln \left (-b x +a \right )}{b}\)	\(27\)
norman	\(\frac {\frac {2 a^{2}}{b}+2 a x}{-b^{2} x^{2}+a^{2}}+\frac {\ln \left (-b x +a \right )}{b}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(-b^2*x^2+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

2*a/b/(-b*x+a)+ln(-b*x+a)/b

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 28, normalized size = 1.08 \begin {gather*} -\frac {2 \, a}{b^{2} x - a b} + \frac {\log \left (b x - a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(-b^2*x^2+a^2)^2,x, algorithm="maxima")

[Out]

-2*a/(b^2*x - a*b) + log(b*x - a)/b

________________________________________________________________________________________

Fricas [A]
time = 2.43, size = 33, normalized size = 1.27 \begin {gather*} \frac {{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{b^{2} x - a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(-b^2*x^2+a^2)^2,x, algorithm="fricas")

[Out]

((b*x - a)*log(b*x - a) - 2*a)/(b^2*x - a*b)

________________________________________________________________________________________

Sympy [A]
time = 0.07, size = 19, normalized size = 0.73 \begin {gather*} - \frac {2 a}{- a b + b^{2} x} + \frac {\log {\left (- a + b x \right )}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/(-b**2*x**2+a**2)**2,x)

[Out]

-2*a/(-a*b + b**2*x) + log(-a + b*x)/b

________________________________________________________________________________________

Giac [A]
time = 1.03, size = 29, normalized size = 1.12 \begin {gather*} \frac {\log \left ({\left | b x - a \right |}\right )}{b} - \frac {2 \, a}{{\left (b x - a\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(-b^2*x^2+a^2)^2,x, algorithm="giac")

[Out]

log(abs(b*x - a))/b - 2*a/((b*x - a)*b)

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 26, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a-b\,x\right )}{b}+\frac {2\,a}{b\,\left (a-b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/(a^2 - b^2*x^2)^2,x)

[Out]

log(a - b*x)/b + (2*a)/(b*(a - b*x))

________________________________________________________________________________________